Hierarchy of Tai mapping for rooted labeled trees · 2013. 7. 12. · Hierarchy of Tai mapping : Time complexity for unordered trees 34 Top-down = Top-down segmental Tai Tree isomorphism

Hierarchy of Tai mapping

for rooted labeled trees

平田耕一(Kouichi Hirata)

(九州工業大学 (Kyushu Institute of Technology))

Contents

2

Tree edit distance

Tai mapping

Variations of Tai mapping


Rooted labeled trees

Rooted tree：a tree with root

Labeled tree：each node is assigned to a label

Ordered trees

Left-to-right order among siblings is given

Unordered trees

Left-to-right order among siblings is not given

Ordered trees vs. unordered trees

Non-isomorphic for ordered trees

Isomorphic for unordered trees

Edit operation

Substitution (relabeling) (sub)

Deletion (del)

Insertion (ins)

Tree edit distance [Tai 79 (J. ACM 26)]

4

ins

sub del

(unordered)

Edit operation

Substitution (relabeling) (sub)

Deletion (del)

Insertion (ins)

Each edit operation is assigned to a cost

Cost

Unit cost：sub=del=ins=1

Indel cost：sub=2，del=ins=1

Tree edit distance

The minimum cost to transform from a tree to another tree

by applying edit operations

Tree edit distance

5

Tree edit distance

6

Unit cost

Ordered tree edit distance = 3

Unordered tree edit distance = 1

del ins sub

sub

Time complexity

Ordered tree edit distance

O(n6) [Tai 79 (J. ACM 26)]

O(n4) [Zhang & Shasha 89 (SIAM J. Comput. 18)]

O(n3logn) [Klien 98 (Proc. ESA’98)]

O(n3) [Demaine et al. 09 (ACM Trans. Algorithms 6)]

n : maximum number of nodes in given two trees

Unordered tree edit distance

NP-hard [Zhang et al. 92 (Inform. Process. Let. 42)]

Unit cost, degree 2 or height 2

MAX SNP-hard [Zhang & Jiang 94 (Inform. Process. Let. 49)]

Indel cost, height 7

Unit cost, degree 2 or height 3 [CPM 12]

7

Tai mapping [Tai 79 (J. ACM 26)]

8

𝑀 ⊆ 𝑇1 × 𝑇2 : Tai mapping

One-to-one

Ancestor condition

Sibling condition (ordered trees)

Tai mapping

The minimum cost of possible Tai mappings

＝tree edit distance

del ins

sub sub

Minimum cost Tai mapping (ordered) Minimum cost Tai mapping (Unordered)

Cost of the Tai mapping

＝3

＝ordered tree edit distance

Cost of the Tai mapping

＝1

＝unordered tree edit distance

Variation of Tai mapping

10

𝑀 ⊆ 𝑇1 × 𝑇2 : Tai mapping, 𝑀− = 𝑀 − {𝑟 𝑇1 , 𝑟 𝑇2 }

Top-down (degree-1) mapping

[Selkow 77 (IPL 6), Chawathe 99 (VLDB)]

∀ 𝑣, 𝑤 ∈ 𝑀− (𝑝 𝑣 , 𝑝 𝑤 ) ∈ 𝑀

LCA-preserving (degree-2) mapping [Zhang 95 (IJFCS 7)]

∀ 𝑣1, 𝑤1 , 𝑣2, 𝑤2 ∈ 𝑀 𝑣1 ⊔ 𝑣2, 𝑤1 ⊔ 𝑤2 ∈ 𝑀

Accordant (Lu’s) mapping [Lu 79 (IEEE PAMI 1), Kuboyama 07]

∀ 𝑣1, 𝑤1 , 𝑣2, 𝑤2 , 𝑣3, 𝑤3 ∈ 𝑀

𝑣1 ⊔ 𝑣2 = 𝑣1 ⊔ 𝑣3 ⇔ 𝑤1 ⊔ 𝑤2 = 𝑤1 ⊔ 𝑤3

𝑟(𝑇) : root of 𝑇 𝑝(𝑣) : parent of 𝑣

𝑣1 ⊔ 𝑣2 : LCA of 𝑣1 and 𝑣2


11

Top-down LCA-preserving

Not top-down

Accordant

Not LCA-preserving

𝑀 : top-down mapping ⇒ 𝑀 : LCA-preserving mapping ⇒ 𝑀 : accordant mapping


12

Constrained (isolated-subtree) mapping

[Zhang 95 (Patten Recog. 28), Zhang 96 (Algorithmica 15)]

∀ 𝑣1, 𝑤1 , 𝑣2, 𝑤2 , 𝑣3, 𝑤3 ∈ 𝑀

𝑣3 < 𝑣1 ⊔ 𝑣2 ⇔ 𝑤3 < 𝑤1 ⊔ 𝑤2

A pair in a constrained mapping is “in a pair of subtrees”

𝑣1 ⊔ 𝑣2 : LCA of 𝑣1 and 𝑣2

Constrained

Not accordant

Not constrained

𝑤1 𝑤2 𝑤3

𝑣1 𝑣2

𝑣3


13

Less-constrained (alignable) mapping

[Lu 01 (COCOON), Kuboyama 07]

∀ 𝑣1, 𝑤1 , 𝑣2, 𝑤2 , 𝑣3, 𝑤3 ∈ 𝑀

𝑣1 ⊔ 𝑣2 < 𝑣1 ⊔ 𝑣3 ⇒ 𝑤2 ⊔ 𝑤3 = 𝑤1 ⊔ 𝑤3

A less-constrained mapping contains no “twist”

A less-constrained mapping

= a mapping for alignment (alignable mapping) [Kuboyama 07]

𝑟(𝑇) : root of 𝑇 𝑣1 ⊔ 𝑣2 : LCA of 𝑣1 and 𝑣2

Not less-constrained Less-constrained

Not constrained

𝑤1 𝑤2 𝑤3

𝑣1 𝑣2

𝑣3 𝑣1 ⊔ 𝑣2

𝑤1 ⊔ 𝑤2

𝑣1 𝑣2

𝑣3 𝑤1

𝑤3 𝑤2


14

Top-down LCA-preserving

Not top-down

Less-constrained

Not constrained

Tai

Not less-constrained

Accordant

Not LCA-preserving

Constrained

Not accordant

𝑀 : top-down mapping ⇒ 𝑀 : LCA-preserving mapping ⇒ 𝑀 : accordant mapping

⇒ 𝑀 : constrained mapping ⇒ 𝑀 : less-constrained mapping ⇒ 𝑀 : Tai mapping


15

Bottom-up mapping [Valiente 01 (SPIRE)]

∀ 𝑣, 𝑤 ∈ 𝑀

∀𝑣′ ∈ 𝑇1[𝑣]∃𝑤′ ∈ 𝑇2[𝑤] 𝑣′, 𝑤′ ∈ 𝑀

∧ ∀𝑤′ ∈ 𝑇2[𝑤] ∃𝑣′ ∈ 𝑇1[𝑣] 𝑣′, 𝑤′ ∈ 𝑀

𝑇[𝑣] : complete subtree of 𝑇 rooted by 𝑣

Bottom-up


Hierarchy of Tai mapping [cf. Kuboyama 07]

16

Top-down

Tai

Tree isomorphism

Bottom-up

Less-constrained

= Alignable

LCA-preserving

Constrained

Accordant Accordant distance

= LCA-preserving distance

if a cost function is a metric


17

Segmental mapping [Kan 12 (ISAAC)]

∀ 𝑣, 𝑤 ∈ 𝑀−

𝑣′, 𝑤′ ∈ 𝑀 ∧ 𝑣′ ∈ 𝑎𝑛𝑐 𝑣 ∧ 𝑤′ ∈ 𝑎𝑛𝑐 𝑤

⇒ 𝑝 𝑣 , 𝑝 𝑤 ∈ 𝑀

𝑎𝑛𝑐(𝑣) : ancestors of 𝑣


18

Top-down segmental mapping [Kan 12 (ISAAC)]

Segmental and 𝑟 𝑇1 , 𝑟 𝑇2 ∈ 𝑀

Bottom-up segmental mapping [Kan 12 (ISAAC)]

∀ 𝑣, 𝑤 ∈ 𝑀

∃ 𝑣′, 𝑤′ ∈ 𝑀 (𝑣 ∈ 𝑎𝑛𝑐 𝑣′ ∧ 𝑤 ∈ 𝑎𝑛𝑐 𝑤′

∧ 𝑣′ ∈ 𝑙𝑣 𝑇1 ∧ 𝑤′ ∈ 𝑙𝑣 𝑇2 )

∨ (𝑣 ∈ 𝑙𝑣 𝑇1 ∧ 𝑤 ∈ 𝑙𝑣 𝑇2 )

𝑙𝑣(𝑇) : set of leaves in 𝑇 𝑎𝑛𝑐(𝑣) : ancestors of 𝑣


19

Bottom-up segmental

Top-down segmental

Not bottom-up

Bottom-up segmental

Not top-down segmental

Not bottom-up

Segmental

Not bottom-up segmental

Not top-down segmental

Hierarchy of Tai mapping [cf. Kuboyama 07]

20

Top-down

Tai

Tree isomorphism

Bottom-up

Less-constrained

= Alignable

LCA-preserving

Constrained

Accordant


21

Segmental

Not LCA-preserving

Not accordant

Not constrained


LCA-preserving

Not segmental

Not bottom-up segmental

Not bottom-up

Hierarchy of Tai mapping [Kan 12 ISAAC]

22

Top-down

= Top-down segmental

Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

LCA-preserving

Constrained

Accordant

Top-down

= Top-down segmental del & ins : leaves

del & ins : leaves or

nodes with one child

del & ins : roots

del & ins : leaves or roots

We can assume that

𝑟 𝑇1 , 𝑟 𝑇2 ∈ 𝑀


23

Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

LCA-preserving

Constrained

Accordant


24

Intersection to alignable mapping

Segmental alignable mapping

Bottom-up segmental alignable mapping

Bottom-up alignable mapping


25

bottom-up mapping

bottom-up alignable mapping


26

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

LCA-preserving

Constrained

Accordant

Hierarchy of Tai mapping [Hirata 13 SIGFPAI88]

27

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

Segmental

alignable

Bottom-up

segmental

alignable

Bottom-up

alignable

LCA-preserving

Constrained

Accordant


28

Intersection to segmental mapping

Constrained segmental mapping

Accordant segmental mapping

LCA-preserving segmental mapping


29

LCA-preserving

LCA-preserving segmental

Not top-down

LCA-preserving

Not LCA-preserving segmental

Not top-down

Segmental alignable

Not constrained segmental

Not accordant segmental

Constrained segmental

Accordant segmental

Not LCA-preserving segmental


30

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

Segmental

alignable

Bottom-up

segmental

alignable

Bottom-up

alignable

LCA-preserving

Constrained

Accordant


31

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

Segmental

alignable

Bottom-up

segmental

alignable

Bottom-up

alignable

LCA-preserving

Constrained

Accordant


= Accordant segmental


Hierarchy of Tai mapping : Metricity

32

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

Segmental

alignable

Bottom-up

segmental

alignable

Bottom-up

alignable

LCA-preserving

Constrained

Accordant



LCA-preserving segmental metric

not metric

Hierarchy of Tai mapping :

Time complexity for ordered trees

33

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

Segmental

alignable

Bottom-up

segmental

alignable

Bottom-up

alignable

LCA-preserving

Constrained

Accordant




𝑂(𝑛𝑚2(1 + log𝑛

𝑚)) = 𝑂 𝑛3 time

𝑂 𝑛𝑚𝐷2 time

≤ 𝑂 𝑛𝑚 time

> 𝑂 𝑛𝑚 time 𝑛 = 𝑇1 𝑚 = |𝑇2| 𝐷 : max. deg.

ISAAC12

ISAAC12

LLLL11

SIGFPAI 90

SIGFPAI 90

𝑂 𝑛𝑚𝐷2 time

SIGFPAI 90

𝑂 𝑛 + 𝑚 time

Hierarchy of Tai mapping :

Time complexity for unordered trees

34

Top-down


Tai

Tree isomorphism

Bottom-up

Segmental

Bottom-up

segmental

Less-constrained

= Alignable

Segmental

alignable

Bottom-up

segmental

alignable

Bottom-up

alignable

LCA-preserving

Constrained

Accordant




LLLL11 tractable

Intractable

(MAX SNP-hard)

𝑂 𝑛𝑚𝑑 time




ALSIP11

ALSIP11

ALSIP11

𝑂 𝑛 + 𝑚 time

𝑛 = 𝑇1 𝑚 = |𝑇2| 𝑑 : min. deg.

𝑂 𝑛𝑚𝑑 time SIGFPAI 90

Edit distance, indel distance: ordered trees

35

Mapping Edit distance Indel distance

Tai 𝑂(𝑛𝑚2(1 + log𝑛

𝑚)) 𝑂(Δ𝐻2 log log 𝑚),

𝑂(𝐿Δ log Δ log log 𝑚)

less-constrained (alignable) 𝑂(𝑛𝑚𝐷2)

constrained (isolated-subtree) 𝑂(𝑛𝑚)

accordant (Lu) 𝑂(𝑛𝑚)

LCA-preserving (degree-2) 𝑂(𝑛𝑚) 𝑂(Δ𝐻 log log 𝑚)

LCA-preserving segmental 𝑂(𝑛𝑚)

top-down (degree-1) 𝑂(𝑛𝑚) 𝑂(𝑛 + 𝑚)

segmental 𝑂(𝑛𝑚)

bottom-up segmental 𝑂(𝑛𝑚)

bottom-up 𝑂(𝑛𝑚) 𝑂(𝑛 + 𝑚)

segmental alignable 𝑂(𝑛𝑚𝐷2)

accordant segmental 𝑂(𝑛𝑚)

[Mozes 09 (TCS 410)]

[Mozes 09 (TCS 410)]

[Valiente 01 (SPIRE)]

[ISAAC 12]

[LLLL 11]

[ISAAC 12]

[SIGPPAI]

[SIGPPAI]

[SIGPPAI]

Δ : sum of pairs of nodes in 𝑇1 and 𝑇2 whose labels are different

Edit distance, indel distance, inclusion:

unordered trees

36

Mapping Edit distance Indel distance Inclusion

Tai MAX SNP-hard MAX SNP-hard NP-complete

less-constrained (alignable) MAX SNP-hard MAX SNP-hard

constrained (isolated-subtree) 𝑂(𝑛𝑚𝑑) 𝑂(𝑛𝑚1.5)

accordant (Lu) 𝑂(𝑛𝑚𝑑) (same as LCA)

LCA-preserving (degree-2) 𝑂(𝑛𝑚𝑑) 𝑂( 𝐷Δ log

2𝑛

𝐷)

𝑂(𝑛𝑚1.5)

LCA-preserving segmental 𝑂(𝑛𝑚𝑑) 𝑂(𝑛𝑚1.5/ log 𝑚)

top-down (degree-1) 𝑂(𝑛𝑚𝑑) 𝑂(𝑛𝑚1.5/ log 𝑚) 𝑂(𝑛𝑚1.5/ log 𝑚)

segmental MAX SNP-hard 𝑂(𝑛𝑚1.5/ log 𝑚)

bottom-up segmental MAX SNP-hard

bottom-up MAX SNP-hard 𝑂(𝑛 + 𝑚) 𝑂(𝑛 + 𝑚)

accordant segmental 𝑂(𝑛𝑚1.5/ log 𝑚)

segmental alignable

Δ : sum of pairs of nodes in 𝑇1 and 𝑇2 whose labels are different

[Shamir & Tsur 99 (J. Algo. 33)]

[Valiente 05 (J. Disc. Algo. 3)]

[Kao 01 (J. Algo. 40)]

[Valiente 01 (SPIRE)]

[LLLL 11]

[ALSIP 12]

[Shamir & Tsur 99 (J. Algo. 33)]

[SIGPPAI]

Future works

37

Improving time complexity computing the following distances

Segmental alignable distance (= segmental alignment)

Alignable distance (= alignment)

Alignment : 𝑂 𝑛𝑚𝐷2 = 𝑂(𝑛4)

Edit distance : 𝑂(𝑛𝑚2(1 + log𝑛

𝑚)) = 𝑂 𝑛3

Analyzing whether computing the following distance is tractable or intractable

Accordant segmental distance

Accordant distance : tractable, segmental distance : intractable

Segmental alignable distance

Segmental distance, alginable distance : intractable

Documents

Hierarchy of Tai mapping for rooted labeled trees · 2013. 7. 12. · Hierarchy of Tai mapping : Time complexity for unordered trees 34 Top-down = Top-down segmental Tai Tree isomorphism